
Research Article


A Survey on Connected Dominating Set Construction Algorithm for Wireless Sensor Networks 

Zhuo Liu,
Bingwen Wang
and
Lejiang Guo



ABSTRACT

In Wireless Sensor Network (WSN), all nodes are energy constrained. Clustering is a kind of energy efficient algorithm, while using a virtual backbone to organize the nodes is a better way. Although, there is no physical backbone infrastructure, a virtual backbone can be formed by constructing a Connected Dominating Set (CDS). The CDS of a graph representing a network has a significant impact on an efficient design of routing algorithms in WSN. A good CDS should first and foremost be small, additionally, it should have other characteristics such as robustness to node failures and low stretch. In this paper, we present a taxonomy and general classification of CDS construction algorithms. We survey different CDS construction algorithms for WSNs.





Received: January 28, 2010;
Accepted: March 15, 2010;
Published: June 10, 2010


INTRODUCTION
Recent advances in miniaturization and lowpower design have led to the development
of smallsized batteryoperated sensors that are capable of detecting ambient
conditions such as temperature and sound. Sensors are generally equipped with
data processing and communication capabilities. The sensing circuitry measures
parameters from the environment surrounding the sensor and transforms them into
an electric signal. Processing such a signal reveals some properties about objects
located and events happening in the vicinity of the sensor. Each sensor has
an onboard radio that can be used to send the collected data to interested parties.
Such technological development has encouraged practitioners to envision aggregating
the limited capabilities of the individual sensors in a large scale network
that can operate unattended (Abbasi and Younis, 2007;
Akyildiz et al., 2002; Chong
and Kumar, 2003. Deborah et al., 1999; Pottie
and Kaiser, 2000; Wang et al., 2003; Rabaey
et al., 2000; Kahn et al., 1999).
Numerous civil and military applications can be leveraged by networked sensors
(Fig. 1). A network of sensors can be employed to gather meteorological
variables such as temperature and pressure.

Fig. 1: 
WSN’s application examples 

Fig. 2: 
Architecture of WSNs 
These measurements can be then used in preparing forecasts or detecting harsh
natural phenomena. In disaster management situations such as earthquakes, sensor
networks can be used to selectively map the affected regions directing emergency
response units to survivors.
In military situations, sensor networks can be used in surveillance missions and can be used to detect moving targets, chemical gases, or the presence of microagents.
Wireless Sensor Networks (WSN) consist a number of wireless nodes. The architecture
of WSN is shown in Fig. 2. The WSNs can be divided into three
parts: data collection networks, basestation and data management center.
In WSN, there is no fixed or predefined infrastructure. Nodes in wireless networks communication via a shared medium, either through a single hop or multihops. Such sharing reduces the network performance due to aggravated ratio interference. At same time it raises energy consumption since packet retransmission is needed when interference occurs. Moreover, the energy consumption could be increased if we don’t limit number of active sensor nodes for the data relays. Backbone can be utilized to address the above problems. Backbone will remove unnecessary transmission links by shutting down some of redundant nodes. Nevertheless backbone will still guarantee network connectivity in order to deliver data efficiently in a wireless sensor network. In virtual backboned based WSNs, some nodes are chosen as dominator node (backbone node) in the backbone construction process.

Fig. 3: 
CDS construction schematic diagram 
All nodes then can directly or indirectly communicate with other nodes via
these dominator nodes. The dominator form the backbone and the nonselected
nodes can perform sleeping schedule or turnoff the ratio to save the energy
consumption. Although, there is no physical backbone infrastructure, a virtual
backbone can be formed by constructing a Connected Dominating Set (CDS). Given
an undirected graph G = (V, E), a subset V'⊆V))fV
is a CDS of G if for each node u ε V, u is either in V' or there exist
a node v ε V' such that uv ε E and the subgraph induced by
V', i.e., G(V'), is connected. The node in the CDS are called dominator (backbone
node), other nodes are called dominatee (nonbackbone node) (Fig.
3). With the help of CDS, routing is easier and can adapt quickly to network
topology changes. To reduces the traffic during communication and simplify the
connectivity management, it is desirable to constructed a minimum CDS. Constructing
a MCDS is proved a NPhard problem (Jeremy et al.,
2004) and in recent years many algorithms of constructing an approximate
MCDS have been proposed.
A good CDS should first and foremost be small, additionally, it should have other characteristics such as robustness to node failures and low stretch (routing in the backbone should not be much longer than the shortest routes over the network topology graph).
In this study, we opt to categorize CDS construction algorithms proposed in the literature for WSNs. We report on the state of the research and summarize their feature and shortcomings. We also compare the different approached.
CDS CONSTRUCTION ALGORITHM COMPARISON
We can compare the CDS construction algorithms in some main aspects. In particular, we have considered the following metrics:
• 
Algorithm duration: The time needed by each protocol
to complete CDS construction 
• 
The overhead per node (in bytes): Associated to the protocol operations.
(These are physical layer measurements, which account for collisions and
for the corresponding automatic packet retransmissions at the MAC layer) 
• 
Energy consumption per node: This metric is important for assessing
the application of CDS construction to networks whose nodes are energy constrained.
If the energy and the overhead cost of maintaining a CDS is high, then CDS
reorganization may impose a nonnegligible burden to the network. On the
other hand, CDS reorganization in terms of gateway rotation is needed to
prevent nodes with more demanding roles from depleting their energy, possibly
impairing network operations 
• 
CDS size: The fraction of the network nodes that are in the CDS
(percentage). A smaller CDS is desirable for minimizing the routing overhead.
In sensor networks with nodes that can turn off their radio interface (energy
saving sleep mode), the CDS size is also a measure of how many nodes need
to stay awake for data transport. Usually, the nodes in the CDS stay up
or have a higher duty cycle for guaranteeing routes to the sink and hence,
the smaller the CDS, the more nodes can be in energy conserving mode 
• 
Over the subgraph G_{b} of the network topology graph induced
by the CDS construction. (G_{b} is the graph where there are no
links between ordinary nodes). This metric gives a measure of how well a
routing protocol can perform over the CDS 
• 
Defined as the number of CDS nodes whose removal (because of failure or
energy depletion), causes CDS disconnection or the uncovering of ordinary
nodes. This is the metric that provides an indirect measure on how long
the network will be operational before requiring a CDS recomputation 
TAXONOMY OF CDS CONSTRUCTION ALGORITHM
From different aspects, CDS construction algorithms can be classified into different types. Here, some classifications are proposed below.
UDG and DGB:
The CDS construction algorithms can classified into two classes, which are Unit
Disk Graph (UDG) based algorithms and Disk Graphs with Bidirectional (DGB) links
(Thai et al., 2007) based algorithms. In UDG
and DGB, the link between any pair of nodes is bidirectional. The nodes transmission
ranges in UDG are the same, while in DGB are different.
The CDS problem has been studied intensively in UDG, in which each node has the same transmission range. The minimum problem in UDG has been shown to be NPhard.
However in practice, the transmission ranges of all nodes are not necessary
equal. In this case, a WSN can be modeled using a directed graph (G = V, E)
(Fig. 4). The nodes in the V are located in a Euclidean plane
and each node v_{i} ε V has a transmission range r_{i}
ε [r_{min}, r_{max}]. A directed edge (v_{i},
v_{j}) ε E if and only if d(v_{i}, v_{j})≤r_{i}
where d(v_{i}, v_{j}) denotes the Euclidean distance between
v_{i} and v_{j}. Such graph are called disk graphs. An edge
(v_{i}, v_{j}) is bidirectional if both (v_{i}, v_{j})
and (v_{j}, v_{i}) are in E, d(v_{i}, v_{j})≤min{r_{i},
r_{j}}. In other words, the edge (v_{i}, v_{j}) is bidirectional
if v_{i} is in the disk D_{j} centered at v_{j} with
radius r_{j} and v_{i} is in the disk D_{i} centered
at v_{i} with radius r_{j}. The minimum CDS problem in DGB is
NPhard since the minimum problem in UDG is NPhard and UDG is a special case
of DGB.
MIS based and NonMIS based:
Independent set (IS(IS⊆V)
is a subset of V. In IS, any pair of nodes is disconnected. Given an Independent
Set (IS), if any one node v(vεV–IS) is added into the IS, there
will be a pair of nodes connected, then the IS is a Maximum Independent Set
(MIS). In MIS, any pair of nodes is disconnected and any node not belonging
to MIS is adjacent to at least one node in MIS.
In both UDG and DGB, the CDS construction algorithms are mainly based on the Maximum Independent Set (MIS).
Li et al. (2005) proposed some distributed connected
dominating set construction algorithms. Li et al.
(2005) proposed a greedy algorithm called SMIS. The SMIS has two phases.
In the first phase, the MIS construction algorithm is described by
Wan et al. (2002). The second phase is the main contribution of Li
et al. (2005) and in this phase, blackblue component is proposed,
which is a connected component of the subgraph induced only by black and blue
nodes and by ignoring connections between blue nodes. The black node is the
dominator of the first phase and the blue node is the dominator of the second
phase. In the second phase, a greedy algorithm is used to find some gray nodes
adjacent to at least 2 black nodes in different blackblue components. The performance
of SMIS is (5.8+ln 4)opt+1.2.
The MIS based algorithms mainly have two kinds of realization. The first one is the two phase based realization, which constructs a MIS at first and then find some optimal nodes to connect the MIS nodes together. The second one is the one phase based realization, which selects the MIS nodes and the optimal intermediate nodes simultaneously. The optimal nodes selection is based on some criterions such as node degree, residual energy of node and node id, etc.
The MIS based algorithms such as Das and Bharghavan (1997),
Raei et al. (2008, 2009),
Rai et al. (2009), Zeng
et al. (2006), Funke et al. (2006),
Yingchang et al. (2009) and Dai
and Wu (2004) are mainly have two stages. The first stage is constructing
a MIS and the second stage is finding some connectors to connect the nodes in
MIS and finally get a CDS. The CDS construction algorithms can have different
criterions. The criterion is the node degree (Das and Bharghavan,
1997; Raei et al., 2008; Rai
et al., 2009). For example, the node degree is the number of the
unmarked neighbors in the first stage and in the second stage, the node degree
is the number of unmarked nodes adjacent to the nonfragment nodes (Das
and Bharghavan, 1997). The algorithm has the performance ratio 2H(Δ)+1
(Das and Bharghavan, 1997). The time and message complexities
are O((n+C)Δ) and O(nC+m+n log n), respectively. The criterion for
constructing a CDS is a timer, which is based on the transmission ranges of
nodes (Raei et al., 2009). The criterion is
the weight of node and the weight is composed of the node degree and the battery
power. The criterion is the node id (Funke et al.,
2006; Yingchang et al., 2009). The node with
the largest id will become a dominator (Funke et al.,
2006), while the node with the smallest id will become a dominator (Yingchang
et al., 2009).
Many algorithms are MIS based. There are also some nonMIS based algorithms
(Dai and Wu, 2004; Rong et al.,
2009).
Wu and Li (1999) and Dai and Wu
(2004) proposed some distributed algorithms in UDG and DGB. A very simple
marking process is proposed by Wu and Li (1999). In the
marking process every node v exchanges its open neighbor set N(v) with its neighbors
and if there exist two unconnected neighbors, v will mark itself. All the marked
nodes form a CDS. The CDS constructed in the first phase contains many redundant
nodes. Wu and Li (1999) proposed two prune algorithms
called Rule1 and 2 to delete some redundant marked nodes. The performance ratio
is not given by Wu and Li (1999), but by Wan
et al. (2002) the performance ratio of this algorithm is proved O(n),
where n is the number of nodes. The message complexity is Θ(m), where m
is the number of edges in the unitdisk graph. The time complexity is O(Δ^{3}),
where Δ is the maximum degree. Dai and Wu (2004)
proposed an extended algorithm for the algorithm. The marking process is the
same as that by Wu and Li (1999), but the pruning process
is improved and denoted by Rulek. Based on the Rulek, the size of the final
constructed CDS is reduced. A hierarchical graph is constructed at first and
then the essential nodes are determinate in the graph according to the Rule
1 (Rong et al., 2009). After the essential nodes
determination phase is ended, there are some redundant dominators, which are
deleted according to the Rule2 in (Rong et al.,
2009). The performance ratio of the algorithm is O(log n) (Rong
et al., 2009).
Centralized algorithm and decentralized algorithm:
Algorithms that construct a CDS can be divided into two categories: centralized
algorithms and decentralized algorithm.
Centralized algorithms:
The centralized algorithms in general yield a smaller CDS with a better performance
ratio than that of decentralized algorithm. Guha and Khuller
(1998) proposed two polynomial time algorithms to construct a CDS in general
graph G. These algorithms are greedy and centralized. The first one has the
approximation ratio of 2(H(Δ)+1) where Δ is the maximum degree of
G and H is a harmonic function. The idea of this algorithm is to build a spanning
tree T rooted at the nodes with maximum degree and grow T until all nodes are
added to T. The nonleaf nodes in T form a CDS. In particular, all nodes in
a given network are white initially. The greedy function that the algorithm
uses to add nodes into T is the number of the white neighbors of each node or
a pair of nodes. The one with the largest such number is marked black and its
neighbors are marked grey. These nodes are then added into T. The algorithm
stops when no white node exists in G. The second algorithm is an improvement
of the first one. This algorithm consists of two phases. The first phase is
to construct a dominating set and the second phase is to connect the dominating
set using a Steinter tree algorithm. With such improved, the second algorithm
has the performance factor of H(Δ)+2. These algorithms later were studied
and implemented by Das et al. (1997), Das
and Bharghavan (1997) and Sivakumar et al. (1998).
Ruan et al. (2004) introduced another centralized
and greedy algorithm of which the approximation ratio is (2+ln Δ).
Decentralized algorithms:
The decentralized algorithms can be further divided into two categories: distributed
algorithms and localized algorithms. In distributed algorithm, the decision
process is decentralized. In the localized algorithm, the decision process is
not only distributed but also requires only a constant number of communication
rounds. Most of the distributed algorithm find a MIS and connect this set. Note
that in an undirected graph, an MIS is also a Dominating Set (DS). Wan
et al. (2002) and Alzoubi et al. (2002a)
proposed a distributed algorithm for a CDS problem in UDG. This algorithm consists
two phases and has the constant approximation ratio of 8. The algorithm first
constructs a spanning tree. Then each node in a tree is examined to find an
MIS for the first phase. All nodes in an MIS are colored black. At the second
phase, more nodes are added (color blue) to connect the black nodes. Later,
Cadei et al. (2002) presented another 2phase
distributed algorithm for a CDS in UDG. This algorithm has the same performance
ratio as the previous one. However, the improvement over (Wan
et al., 2002) is the message complexity. The root does not need to
wait for the complete message from the furthest nodes. Recently, Li
et al. (2005) proposed another distributed algorithm with
a better approximation ratio, which is (4.8+ln 5). This algorithm also has two
phases. At the first phase, an MIS is found. At the second phase, a Steiner
tree algorithm is used to connect the MIS.
For the localized algorithms, Wu and Li (1999) proposed
a simple algorithm that can quickly determine a CDS based on the connectivity
information within the 2hops neighbors. This approach uses a marking process.
In particular, each node is marked true if it has two unconnected neighbors.
All the marked nodes form a CDS. The researchers also introduced some dominant
pruning rules to reduce the size of the CDS. Wan et al.
(2002) showed that the performance ratio of Wu and Li
(1999) is within a factor of O(n) where n is the number of nodes in a network.
Taxonomy of CDS construction algorithm is shown in Fig. 5.

Fig. 5: 
Taxonomy of CDS construction algorithm 
Alzoubi et al. (2002b) proposed another localized
2phase algorithms with the performance ratio of 192. At the first phase, an
MIS is constructed using the 2hops neighbors information. Specifically, once
a node knows that it has the smallest ID within its neighbors, it becomes a
dominator. At the second phase, the dominators are responsible for identifying
a path to connect the MIS. Li et al. (2004) proposed
another localized algorithm with the performance ratio of 172. This localized
algorithm has only 1 phase. A node marks itself as a dominator if it can cover
the most white nodes compared to its 2hops neighbors.
CDS CONSTRUCTION ALGORITHMS FOR WSNs
Generally, WSNs involve a large number of sensors ranging in the hundreds or even thousands. The CDS construction is an effective mean for managing such high population of nodes. Here, we present a literature survey of published algorithms for constructing CDS in WSN. Following the increased interest in wireless sensor networks, many distributed approaches have been proposed because of no requirements for global network topology knowledge.
Distributed approximation algorithms for MCDS in WSN were first developed in
a series of papers (Das and Bharghavan, 1997; Sivakumar
et al., 1998). These algorithms provided distributed implementations
of the greedy heuristics (Guha and Khuller, 1998). The
CDS is referred to in these papers as a spine and functions as a virtual backbone.
The primary task of the spine is the route computation and maintenance. Nodes
in the spine maintain up to date information about their domains (neighbors
of the CDS nodes) and are used to exchange such information between each other
to store global information of the network. The advantage of this strategy is
the storage of global information in fewer nodes than the num number of nodes
in the network, which reduces the access overhead for this information and reduces
the update overhead. According to this strategy, it is highly desirable to reduce
the size of the CDS in order to minimize the access and update overhead. The
spine is not necessarily used to route packets in the network, even though it
can be used to provide a temporary backup routes for fault tolerance.
We notice that these algorithms lack mechanisms to bridge two consecutive stages. Specifically, individual nodes have no way to tell when the next stage should begin. Furthermore, these approximation algorithms suffer from high performance ratio and high implementation complexities, O(n) time and messages. The construction of the minimum CDS requires 2hop neighborhood knowledge, which means larger message size, frequent updates, slower convergence speed and more memory.
Alzoubi et al. (2002c) proposed two famous distributed heuristics algorithms: (1) IDbased approach for CDS and (2) levelbased approach for CDS.
IDbased approach:
The construction of the CDS uses the MIS generated by the IDBased approach
for rank assignment. A variation of this algorithm was proposed by Alzoubi
et al. (2002b), but the levelbased approach was used for rank assignment.
Thus, the performance ratio was 8 instead of 12. Here, the distributed algorithm
for CDS consists of three procedures
• 
Leader Election 
• 
MIS construction 
• 
Dominating tree construction 
The leader election procedure elects a node e.g., with the smallest ID, as
the leader. The distributed algorithm in (Cidon and Mokryn,
1998) for leader election can be adopted. This algorithm has a message complexity
of O(n log n). When the leader is found, it broadcasts its identity to all the
nodes in the network.
When a node receives the leader’s identity, it starts the MIS construction procedure, which is described earlier and using the IDbased approach for rank assignment. Each node will either be colored with black (as a dominator) or gray (as a dominatee). The leader will also select a black node as the root of the dominating tree as follows: If the leader is marked black, it selects itself as the root of the tree, otherwise it selects one of the black neighbors to be the root.
The dominating tree construction procedure is initiated by the root to construct a tree containing all black nodes in addition to some gray nodes. All nodes in this dominating tree form a CDS. The root joins the dominating tree first, then it sends an invitation message to all black nodes within 3hop distance to join the tree. When a node receives an invitation message and all its neighbors have been marked either black or gray, it responds to the message. When each black node joins the dominating tree, it will also send an invitation to all black nodes within 3hop distance to join the tree. This invitation will be relayed through the gray nodes within 2hops distance. Each black node will join the tree when it receives the invitation for the first time together with the gray nodes which relays the invitation to itself. This process should be repeated until all black nodes are in the tree.
Result:
All nodes in the dominating tree form a CDS with size at most 12opt+3. In addition, the algorithm has O(n log n) message complexity and O(n) time complexity.
Levelbased approach:
The construction of the CDS uses the MIS generated by the levelbased approach
for rank assignment. This algorithm can be divided into three phases: The leader
election phase, the level calculation phase and the color marking phase. The
distributed algorithm for leader election can be adopted by Cidon
and Mokryn (1998). This algorithm has O(n log n) message complexity and
O(n) time complexity. After the construction of the spanning tree is completed
and when the root receives levelcomplete messages from all its children, each
node knows the level numbers of all its neighbors. The pair (level, ID) of a
node defines the rank of this node. The ranks of all nodes are sorted in the
lexicographic order. Thus the leader, which is at level 0, has the lowest rank.
In the color marking phase all nodes are initially unmarked (white) and will
eventually get marked either black or gray. Two types of messages are used by
the nodes during this phase, the dominator message and the dominatee message.
The dominator message is sent by a node after it marks itself black and the
dominatee message is sent by a node after it marks itself gray. Both messages
contain the sender’s ID. The algorithm can be described as a color markup
process. Initially, the root marks itself black and then broadcasts to its neighbors
a dominator message. All other nodes act according to the following principles.
• 
Whenever a white node receives a dominator message from a
white neighbor for the first time, it marks itself gray and
broadcasts the dominatee message 

Fig. 6: 
State transition diagram 
• 
When a white node has received a dominatee message from each
of its neighbors of lower rank, it marks itself black and broadcasts
the dominator message 
• 
When a gray node receives a dominator message for the first time
from one of its children in T, which has never sent a dominatee message,
it remarks itself black and broadcasts the dominator message 
Figure 6 shows the state transition diagram of this phase.
Eventually each node will be either black (a dominator) or gray (dominatee).
A reporting process, if necessary, can be performed to notify the root of the
completion. When a leaf node has determined its status, it transmits a report
complete message to its parent. Each internal node will wait till it receives
this report complete message from each of its children and then forward it up
the tree to the root. When the root receives the report complete message from
all its children then the tree is completed.
Result:
All nodes form a CDS with size at most 8opt+1. In addition, the message complexity of the algorithm is O(n log n) and the time complexity is O(n).
Das et al.’s algorithm:
The centralized version of the distributed algorithm proposed by Das
et al. (1997) consists of three stages. The first stage finds an
approximation to minimum dominating set, which is essentially the wellstudied
set cover problem. Not surprisingly, the heuristic proposed by Das
et al. (1997), Bharghavan and Das (1999),
Das and Bharghavan (1997) and Sivakumar
et al. (1998) is a translation of Chvatal’s greedy algorithm
(Chvátal, 1979) for set cover and thus guarantees
an approximation factor of H(Δ), where Δ is the maximum degree and
H is the harmonic function. Let U denote the dominating set output in this stage.
The second stage constructs a spanning forest F. Each tree component in F is
a union of stars centered at the nodes in U.

Fig. 7: 
Instance for which the size of solution output by Das et
al.’s algorithm 
The stars are generated by letting each dominatee node pick up an arbitrary
neighbor in U. The third stage expands the spanning forest F to a spanning tree
T. All internal nodes in T form a CDS. It is easy to show that the CDS generated
in this way contains at most 3U nodes and therefore is a 3H(Δ) approximation
of minimum CDS.
Figure 7 shows a family of instances for which the size of
the solution computed by the above greedy algorithms is larger than the optimum
solution by a logarithm factor. All points lie in a rectangle whose horizontal
side has length one and whose vertical side has length (the
selection of the value is
based on the fact for any two points whose ycoordinates are differed by ,
they are connected if and only if their xcoordinates are differed by
at most 1/(2(k–1)). The two nodes v_{1} and v_{k} are
the centers of the left and right vertical sides, respectively. The k–2
nodes v_{2}, v_{3},...,v_{k–1} are evenly distributed
within the line segment between and from left to right. The two nodes u_{1}
and u_{2} are the centers of the two subrectangles above and below
the segment between v_{1} and v_{k}, respectively. The rest
points lie in the two horizontal sides. In each horizontal side, 2^{0}
= 1 node lies to the left of (and excluding) the perpendicular bisector of v_{1}v_{2},
2^{k–1} nodes lie to the right of the perpendicular bisector of
v_{k–1}v_{k} and 2^{i–1} nodes lie between
the perpendicular bisector of v_{i–1}v_{i} and the perpendicular
bisector of v_{i}v_{i+1}. Thus, the total number of nodes is:
where, u_{1} is adjacent to all nodes lying in the top subrectangle,
u_{2} is adjacent to all nodes lying in the bottom subrectangle and
they are adjacent to each other. Thus, {u_{1}, u_{2}} forms
a minimum CDS.
Result:
All nodes form a CDS with size at most 3H(Δ). In addition, the message complexity of the algorithm is O(n^{2}) and the time complexity is O(n^{2}).
Wu and Li’s algorithm:
While the algorithm proposed by Das et al. first finds a DS and then
grow this DS into a CDS, the algorithm proposed by Wu and
Li (1999) takes an opposite approach. The algorithm in Wu
and Li (1999) first finds a CDS and then prune certain redundant nodes from
the CDS. The initial CDS U consists of all nodes which have at least two nonadjacent
neighbors. A node u in U is considered as locally redundant if it has either
a neighbor in U with larger ID which dominates all other neighbors of u or two
adjacent neighbors with larger IDs which together dominate all other neighbors
of u. The algorithm then removes all locally redundant nodes from U. This algorithm
applies only to wireless ad hoc networks whose unitdisk graph is not a complete
graph. As indicated by Wu and Li (1999), the approximation
factor of this algorithm remains unspecified. Obviously, the MCDS of any wireless
ad hoc network whose unitdisk graph is not complete graph consists of at least
two nodes. On the other hand, any CDS contains at most n nodes. Thus, the approximation
factor of the above algorithm is at most n/2 where n is the number of nodes.
Next, we show that the approximation factor of the above algorithm is exactly
n/2. This means that the above algorithm does perform extremely poorly over
certain instances.
When n is even, we consider the instance illustrated in Fig.
8a. These nodes are evenly distributed over the two horizontal sides of
a unitsquare. Each node has exactly m neighbors, one in the opposite horizontal
side and the rest in the same horizontal side. The two nodes at the left two
corners of the unitsquare form an minimum CDS. However, the CDS output by the
algorithm in (Wu and Li, 1999) consists of all nodes.
Indeed, for each node u, the unique neighbor lying in the opposite horizontal
side is not adjacent to all other neighbors of u. Thus, the initial CDS U consists
of all nodes. In addition, no single neighbor of a node u can dominate all other
neighbors of u. Furthermore, if a pair of neighbors of u is adjacent, they must
lie in the same horizontal side as u and therefore neither of them is adjacent
to the unique neighbor of u lying in the opposite horizontal side. So, no node
is locally redundant. Consequently the output CDS still consists of all nodes.

Fig. 8: 
Instance for which the CDS output by Wu and Li’s algorithm 
When n is odd, we consider the instance illustrated in Fig. 8b.
The node with the largest ID, denoted by u*, is the center of the left vertical
side of a unitsquare and all other n–1 nodes are evenly distributed over
the two horizontal sides of the unitsquare. The two nodes at the left two corners
of the unitsquare form a minimum CDS. On the other hand, the CDS output by
the algorithm by Wu and Li (1999) also consists of all
nodes. In fact, following the same argument as in the even case, all nodes other
than u* are in the initial CDS U. The node u* is also in the initial CDS U as
well. Since, u* is not adjacent to the two nodes at the right corners of the
unitsquare, all nodes other than u* are not locally redundant. The u* itself
is also not locally redundant as it has the maximum ID. Therefore, the output
CDS still consists of all nodes.
Result:
All nodes form a CDS with size at most n/2. In addition, the message complexity of the algorithm is Θ(m) and the time complexity is o(Δ^{3}).
Stojmenovic et al.’s algorithm:
In the context of clustering and broadcasting, Stojmenovic
et al. (2001) presented three synchronized distributed constructions
of CDS. In each of the three constructions, the CDS consists of two types of
nodes: the clusterheads and the bordernodes. The clusterheads form a Maximal
Independent Set (MIS), i.e., a dominating set in which any pair nodes are nonadjacent.
A node is a bordernode if it is not a clusterhead and there are at least two
clusterheads within its 2hop neighborhood. The set of clusterheads is induced
by a ranking of nodes which give rise to a total ordering of all nodes. Three
rankings are used: the ID only (Gerla et al., 1995;
Lin and Gerla, 1997), an ordered pair of degree and ID (Chen
and Stojmenovic, 1999) and an order pair of degree and location (Stojmenovic
et al., 2001). The selection of the clusterheads is given by a synchronized
distributed algorithm, which can be generalized to the following framework.
Initially all nodes are colored white. In each stage of the synchronized distributed algorithm, all white nodes which have the lowest rank among all white neighbors are colored black; then all white nodes adjacent to the these black nodes are colored gray; finally the ranks of the remaining white nodes are updated. The algorithm ends when all nodes are colored either black or gray. All black nodes then form the clusterheads.
Regardless of the choice of the ranking, the algorithms by Stojmenovic
et al. (2001) have a Θ(n) approximation factor. Such inefficiency
stems from the nonselective inclusion of all ordernodes. In fact, if the rank
is ID only, Fig. 9 shows a family of instances which would
imply the approximation factor to be exactly n, the worst possible. In these
instances, the node with the largest ID is located at the center of a unitdisk
and all other nodes are evenly distributed in the boundary of the unitdisk.
After the clusterheads are selected, all other nodes become bordernodes. Thus
the CDS would consist of all nodes. On the other hand, the node at the center
dominates all other nodes. If the rank is an ordered pair of degree and ID or
an order pair of degree and location, the instances shown in Fig.
8 imply that their approximation factors are at least n/2.
All algorithms by Stojmenovic et al. (2001)
have o(n^{2}) message complexity and Ω(n) time complexity. This
can be illustrated in the following instance: All n nodes are evenly distributed
in an interval of length n±1 with two nodes being the endpoints of the
interval. The ith node from the left endpoint of interval has ID i.
Result:
All nodes form a CDS with size at most n. In addition, the message complexity of the algorithm is O(n^{2}) and the time complexity is Ω(n).

Fig. 9: 
Instance for which the CDS output by Stojmenovic et al.’s
algorithm 
Zhuo Liu et al.’s algorithm:
Liu et al. (2010a, b) proposed
a good performance CDS construction algorithm with small size for WSNs. The
algorithm has three stages. The first stage is constructing a connected set
CS (connected set) and the second stage is constructing a connected dominating
set CDS and the third stage is pruning the redundant dominators of CDS.
We compare the size of CDS constructed by ATISA with four algorithms: Wu’s
algorithm (Wu and Li, 1999) (denoted by WUA), Wang’s
algorithm (Wan et al., 2002) (denoted by WAA),
the first algorithm of Xiang’s algorithms by Yingchang
et al. (2009) (denoted by XA) and the first algorithm of Thai’s
algorithms by Thai et al. (2007) (denoted by
TFA) (Fig. 10).
Result:
All nodes form a CDS with size at most (9.67+19n_{k})opt. In addition, the message complexity of the algorithm is O(n) and the time complexity is O(n).
Table 1: 
Performance comparison 


Fig. 10: 
The size of the CDS 
We have surveyed some famous CDS construction algorithms. Performance comparison
of more algorithms is shown in Table 1.
CONCLUSIONS
The CDS have proven to be an effective construct within which to solve a variety of problems that arise in wireless networks. Significant attention has been paid to CDS construction algorithms yielding a large number of publications. In this study, we surveyed and classified the CDS construction algorithms and classified.
As WSN moves from the research labs to the real world, additional requirements will certainly arise. Issues that will likely arise include the security and survivability of CDSbased applications. In infrastructurebased wireless networks, the infrastructure plays a key role for security purposes. A perimeter is established around the infrastructure, regulating access to the network. A CDS could perform similar actions through forming a virtual network infrastructure. However, significant issues must first be addressed including the realtime assurance of the trust worthiness of mobile hosts. Similarly, performance of the CDS construction algorithms in the face of security attacks must be evaluated in order to ensure robust, survivable network functionality.
ACKNOWLEDGMENTS
This study is supported by two grants from the National Natural Science Foundation of China (No. 60773190, No. 60802002)

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